Hyperbolic geometry and mapping class groups

University Of Utah, Salt Lake City UT

Investigators

Abstract

Surfaces and 3-manifolds are important objects of study not only because of the rich mathematics they contain but also because they are fundamental objects that appear in the real world. One approach for studying surfaces and 3-manifolds is to give them geometry. When doing so one is naturally lead to hyperbolic geometry because as soon as one looks past the simplest examples the geometry of "most" surfaces and 3-manifolds is hyperbolic. Hyperbolic geometry is one of the three classical geometries. The other two are the more familiar Euclidean, or flat, geometry of a table top or piece of paper and the spherical geometry of a tennis ball or the earth. Hyperbolic geometry, while less familiar, locally is modeled on the flaring end of a trumpet or trombone and it is hyperbolic geometry that is at the center of the research in this proposal. This project studies several related topics. The projection complex is a construction developed by the PI in collaboration with M. Bestvina and K. Fujiwara. It was originally used to study the mapping class group but has shown to have applications to other groups as well. One aspect of this project is to continue the study of this object and its applications. The other main focus of the project is the study of hyperbolic 3-manifolds. In particular the PI will continue, in collaboration with J. Brock, R. Canary and C. Lecuire, the study of the topology of deformation spaces of hyperbolic 3-manifolds.